Problem: Simplify and expand the following expression: $ \dfrac{2t - 4}{4t + 3}-\dfrac{t - 9}{2t - 9} $
In order to subtract expressions, they must have a common denominator. Get both fractions over a common denominator of $(4t + 3)(2t - 9)$ Multiply the first term by $\dfrac{2t - 9}{2t - 9}$ $ \begin{align*} \dfrac{2t - 4}{4t + 3} \times \dfrac{2t - 9}{2t - 9} & = \dfrac{(2t - 4)(2t - 9)}{(4t + 3)(2t - 9)} \\ & = \dfrac{4t^2 - 26t + 36}{(4t + 3)(2t - 9)}\end{align*} $ Multiply the second term by $\dfrac{4t + 3}{4t + 3}$ $ \begin{align*} \dfrac{t - 9}{2t - 9} \times \dfrac{4t + 3}{4t + 3} & = \dfrac{(t - 9)(4t + 3)}{(2t - 9)(4t + 3)} \\ & = \dfrac{4t^2 - 33t - 27}{(2t - 9)(4t + 3)}\end{align*} $ Now we have: $ = \dfrac{4t^2 - 26t + 36}{(4t + 3)(2t - 9)} - \dfrac{4t^2 - 33t - 27}{(2t - 9)(4t + 3)} $ Now both terms have a common denominator we can subtract the numerators: $ = \dfrac{4t^2 - 26t + 36 - (4t^2 - 33t - 27)}{(4t + 3)(2t - 9)} $ $ = \dfrac{4t^2 - 26t + 36 - 4t^2 + 33t + 27}{(4t + 3)(2t - 9)} $ $ = \dfrac{7t + 63}{(4t + 3)(2t - 9)}$ Expand the denominator: $ = \dfrac{7t + 63}{8t^2 - 30t - 27}$